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AC Stark Effect, Born-Oppenheimer Approximation, Molecular Quantum Mechanics, Franck-Condon Factors
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\documentclass[12pt,fleqn]{article}
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\begin{document}

\title{The AC Stark Effect, Time--Dependent Born--Oppenheimer
Approximation, and Franck--Condon Factors}
\author{George A. Hagedorn\thanks{Partially
Supported by National Science Foundation
Grant DMS--0303586.}\\
Department of Mathematics and\\
Center for Statistical Mechanics and Mathematical Physics\\
Virginia Polytechnic Institute and State University\\
Blacksburg, Virginia 24061-0123, U.S.A.\\[10pt]
\and
Vidian Rousse\\
D\'epartement de Math\'ematiques\\
Universit\'e de Cergy--Pontoise\\
2 Avenue Adolphe Chauvin\\
BP 222, Pontoise\\
95 302 Cergy--Pontoise Cedex, France\\[10pt]
\and
Steven W. Jilcott, Jr.\\
BAE Systems Advanced Information Technology\\
6 New England Executive Park\\
Burlington, MA 01803, U.S.A.}

\maketitle
\begin{abstract}
We study the quantum mechanics of
a simple molecular system that is subject
to a laser pulse. We model the laser pulse
by a classical oscillatory electric field,
and we employ the Born--Oppenheimer approximation
for the molecule.
We compute transition amplitudes to leading order
in the laser strength. These amplitudes
contain Franck--Condon factors
that we compute explicitly to leading order
in the Born--Oppenheimer parameter.

We also correct an erroneous calculation
in the mathematical literature on the
AC Stark effect for molecular systems.
\end{abstract}

\newpage
\baselineskip=21pt
\parskip 3mm

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}\label{intro}
%\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In many experimental situations, laser pulses
are used to stimulate electronic transitions in molecules.
After such a transition, a molecule is typically in
a superposition of many vibrational states for the
nuclei.
The main goal of this paper is to study these transitions
in a simple molecular model, and to calculate the
particular superposition of vibrational states.

In the physics literature, problems of this type have
a long history, with intuition and formulas that date back
to the early days of quantum mechanics. The results rely
on two approximations. The first is the
Born--Oppenheimer approximation, which separates the
nuclear and electronic motions. In this approximation,
the electronic bound state energy levels
depend on the nuclear positions, and these electronic
energy levels play the role of effective potentials for
the nuclei. This approximation depends on the smallness
of the parameter $\eps$, which is the fourth root
of the electron mass divided by the mean nuclear mass.
The second approximation involves
the idea that electronic transitions occur on a time scale
that is very short compared with that of the nuclear motion.
Often, the term ``vertical transition'' is used.
%
\begin{figure}[h]
\begin{center}
\scalebox{0.59}{\input{figureforGeorge.pstex_t}}
\caption{Graphical interpretation of a vertical transition
\label{ourfigure}}
\end{center}
\end{figure}
%
This terminology comes
from plotting the nuclear positions on a horizontal axis
and electron energy levels on the vertical axis, as in
Figure \ref{ourfigure}. If the
nuclei do not move during a transition, then a change of
electronic state corresponds just to a vertical change
on the plot.
Quantitatively, this idea corresponds to the nuclear
wave packet not changing during the time that
the electrons are making the transition.



Although the molecule may start in a single bound state
before the electronic transition, it typically is not in a
single bound state after the transition because the nuclei
find themselves subject to a different effective potential.
However, if the nuclei are in a well of the new
electron energy level, the final nuclear state can be
expanded as a superposition of bound states of the new well.
The expansion coefficients are called the Franck--Condon
factors \cite{Condon1,Condon2,Franck}.

We prove that this intuition is correct to
first order in the laser field strength $\mu$, and
we present a formula for the Franck--Condon factors to
leading order in $\eps$. For technical reasons, we do
this for a situation where the nuclei have only one degree
of freedom. This model is specifically relevant to diatomic
molecules where the single degree of freedom is the distance
between the nuclei.

In addition, we correct some calculations in
\cite{Jilcott1,Jilcott2}. Those articles study the
time--dependent Born--Oppenheimer approximation in the
presence of a laser pulse. They develop expansions jointly
in $\eps$ and $\mu$ for the full molecular wave function.
The error we correct stems from the inadvertent commuting
of two operators that do not commute (in the middle of
a long calculation). The error is largely hidden in the
complicated notation that the problem appears to require.

The paper is organized as follows:\ \, In Section 2,
we make a precise statement of the problem and introduce
the notation we require. In Section 3, we study electronic
transitions to leading order in $\eps$ and first order in
$\mu$. We prove that the intuition described above is
correct, and we present the expression
for the Franck--Condon factors the leading order in $\eps$.
Our main result is summarized in Theorem \ref{main}.
In Section 4, we indicate how to correct the calculational
error in \cite{Jilcott1,Jilcott2}.

\vskip 3mm \noindent
{\bf Acknowledgement.} George Hagedorn would like to
thank the Universit\'e de Cergy--Pontoise for its
hospitality and support during June 2005 when this paper
was written.
Vidian Rousse would like to thank Virginia Polytechnic
Institute and State University for its hospitality and
support during April 2004 where the main idea for this
paper was initiated.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Precise Formulation of the Problem}\label{section2}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Our goal is to study approximate solutions
to the molecular Schr\"odinger equation
\be\label{SE}
i\,\eps^2\,\frac{\partial\Psi}{\partial t}\ =\
-\,\frac{\eps^4}2\,\frac{\partial^2\Psi}{\partial x^2}\,
+\,h(x)\,\Psi\,+\,\mu\,X\,f(t,\,\eps)\,\Psi.
\ee
for small $\eps$ and small $\mu$.
Here $\Psi$ is a function of the nuclear configuration $x$,
and takes its values in a Hilbert space
${\cal H}_{\mbox{\scriptsize el}}$
that we call the electronic Hilbert space. The operator
$h(x)$ is called the electronic Hamiltonian. It is
self--adjoint on ${\cal H}_{\mbox{\scriptsize el}}$ and
depends smoothly on $x$ in a strong resolvent sense.
Discrete eigenvalues of $h(x)$ that depend smoothly on $x$
are called electron energy levels.

We assume $h(x)$ has two non--degenerate eigenvalues
$E_{\w}(x)$ and $E_{\z}(x)$ that are each isolated from the rest
of the spectrum and from one another.
We denote the corresponding normalized electron eigenvectors
by $\Phi_{\w}(x)$ and $\Phi_{\z}(x)$.
We choose the phases of these eigenvectors so that they each
satisfy
$\langle\,\Phi(x),\,\Phi'(x)\,
\rangle_{{\cal H}_{\mbox{\scriptsize el}}}
=0$.
If they can be chosen to be real vectors
in some $L^2$ space, this phase condition
is automatically satisfied.

We assume $E_{\w}$ has local minimum
at $x=a_{\w}$ with $E_{\w}''(a_{\w})>0$.
We further assume that $x=a_{\w}$ lies inside a well of $E_{\z}$.
We require either that $a_{\w}$ be a local minimum of $E_{\z}$
with $E_{\z}''(a_{\w})>0$ or
that both $E_{\z}'(a_{\w})\neq 0$
and that within that well, any local maxima of $E_{\z}$ lie
strictly below $E_{\z}(a_{\w})$. So that we are considering bound
states instead of resonances, we assume $E_{\w}(a_{\w})$ and
$E_{\z}(a_{\w})$ lie strictly below
$\displaystyle \liminf_{|x|\to\infty}\,\inf\,\sigma(h(x))$.

For most of our arguments, the specific form of $f$
is not essential. However,
to model realistic laser pulses,
we can take $f(t,\,\eps)$ to have the form
$$f(t,\,\eps)\ =\ \eps^{-q}\ f_0\left(\,t/\eps^{2-q}\,\right)\
\cos\left(\,\omega\,t/\eps^2\,\right),$$
with $0\le q\le 2$.
A reasonable choice of $f_0$ is a non-negative smooth
function of compact support that specifies an envelope
for the pulse.

For simplicity, we assume that $X$ is a bounded operator
on the electron Hilbert space. The more general
situation where $X$ may be unbounded and may also act
on the nuclei can be handled as in \cite{Jilcott1,Jilcott2}.
Because of
the large masses of the nuclei as $\eps\rightarrow 0$,
and because the laser field oscillates very rapidly on
the time scale of the nuclei, inclusion of the action
of $X$ on the nuclei has no effect to leading order
in $\eps$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Born--Oppenheimer Preliminaries}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we collect several known results
that we need. These concern semiclassical wave packets
and both the time--dependent and time--independent
Born--Oppenheimer approximations. More detailed discussions
of these results can be found in the references we have
cited.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Semiclassical Wave Packets}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The time--dependent Born--Oppenheimer approximation makes
use of certain semiclassical wave packets for the nuclei.
In one dimension, these may be written as
\bea\label{gaussians1D}
&&\hskip -2cm \phi_m(A,B,\eps^2,a,\eta,x)\\[3mm] \nonumber
&=&
2^{-m/2}\,(m!)^{-1/2}\,
\left(\,\overline{A}/A\,\right)^{m/2}\
H_m\left(\,\hbar^{-1/2}|A|^{-1}(x-a)\right)\
\phi_0(A,B,\eps^2,a,\eta,x),
\eea
where $m=0,\,1,\,2,\,\cdots,$ and
$$\phi_0(A,B,\eps^2,a,\eta,x)\ =\
\pi^{-1/4}\,\eps^{-1/2}\,A^{-1/2}\
\exp\left(\ -\ \frac{B\,(x-a)^2}{2\,A\,\eps^2}\ +\ i\
\frac{\eta\,(x-a)}{\eps^2}\ \right).$$
We always assume $A$ and $B$ are chosen to satisfy
$\mbox{Re}\,(\overline{B}\,A)\ =\ 1$.
Each of these states $\phi_m$
is concentrated near the
classical position $a$ and its (scaled) Fourier transform
is concentrated near the classical momentum $\eta$.
%GH
%Next two lines \eps, not \eps^2
%
The uncertainty in position is $\eps\,\sqrt{m+1/2}\ |A|$
and its uncertainty in momentum is $\eps\,\sqrt{m+1/2}\ |B|$.
For fixed $a$, $\eta$, $A$, and $B$, these functions
form an orthonormal basis of $L^2(\R)$, indexed by $m$.
Numerous properties of these semiclassical wave packets
and  their higher dimensional analogs are discussed in
\cite{raise}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Lowlying Vibrational States}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We take our initial conditions for (\ref{SE}) to be the
$m^{\mbox{\scriptsize th}}$ vibrational state associated
with the electronic level $E_{\w}$.
From known results of the time--independent
Born--Oppenheimer approximation \cite{tibo},
this state equals
\be\label{initcond}
\phi_m(A_{\w},B_{\w},\eps^2,a_{\w},0,x)\,\Phi_{\w}(x)
\ +\ O(\eps),
\ee
where\quad $A_{\w}=1/\omega_{\w}$\quad and\quad
$B_{\w}=\omega_{\w}$\quad
with\quad $\omega_{\w}\,=\,\sqrt{E_{\w}''(a_{\w})}$.

The energy of this state is\quad
$E_{{\w},m}\ =\ E_{\w}(a_{\w})
\ +\ \eps^2\,(m+1/2)\,\omega_{\w}\ +\ O(\eps^4)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{The Leading Order Time--Dependent
Born--Oppenheimer Approximation}\label{BS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Except in Section \ref{jilcott},
we only require the leading order
time--dependent Born--Op\-pen\-heim\-er approximation
\cite{tdbo,texas,memoir}.
For any given isolated non--degenerate electron
level $E_{\zz}(x)$ with normalized eigenfunction
$\Phi_{\zz}(x)$
with $\langle\,\Phi_{\zz}(x),\,\Phi_{\zz}'(x)\,
\rangle_{{\cal H}_{\mbox{\scriptsize el}}}=0$,
equation (\ref{SE}) with $\mu=0$
has solutions of the form
\be\label{LowestTDBO}
e^{iS(t)/\eps^2}\
\phi_m(A(t),B(t),\eps^2,a(t),\eta(t),x)\
\Phi_{\zz}(x)\ +\ O(\eps),
\ee
where
\bea\label{adot}
\dot{a}(t)&=&\eta(t),\\[2mm]\label{etadot}
\dot{\eta}(t)&=&-\ E_{\zz}'(a(t)),\\[2mm]\label{Adot}
\dot{A}(t)&=&i\,B(t),\\[2mm]\label{Bdot}
\dot{B}(t)&=&i\,E_{\zz}''(a(t))\,A(t),
\qquad\mbox{and}\\[2mm]\label{Sdot}
\dot{S}(t)&=&\frac{\eta(t)^2}2\ -\ E_{\zz}(a(t)).
\eea
We use the symbol $U_{BO}(t,\,s)$ to indicate
the lowest order Born--Oppenheimer propagation,
which agrees with the exact propagator
up to an $O(\eps)$ error when applied to any
semiclassical wave packet in $x$
times a discrete eigenstate of $h(x)$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Molecular Bound States Near a Fixed
Energy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Although our initial state is the
$m^{\mbox{\scriptsize th}}$
vibrational state of the well for $E_{\w}$,
the component of the wave function that has made
an electronic transition is generically in
a superposition of highly
excited vibrational states for the well $E_{\z}$.
The energy of these excited states is near
$E_{\z}(a_{\w})$,
but the vibrational levels have energy spacing
$O(\eps^2)$, so a relevant vibrational level
is the $n^{\mbox{\scriptsize th}}$ one, where
$n=O(\eps^{-2})$.
These states are rigorously studied as
quasimodes in \cite{rousse2,rousse3}.
The $n^{\mbox{\scriptsize th}}$ quasimode is
$\psi_{n,\eps}(x)\,\Phi_{\z}(x)\,+\,O(\eps)$,
where $\psi_{n,\eps}$ can be written as an integral over
a classical trajectory
for the potential $E_{\z}$ to leading order in $\eps$.
To avoid confusion with the time $t$ in our problem,
we parameterize the orbit by the time variable $r$.
More precisely,
\bea\label{psin}\hskip -27pt
\psi_{n,\eps}(x)&=&\pi^{-1/4}\,\eps^{-1/2}\,
\sqrt{\frac{|\theta|}{2\,\tau(E)}}\\[4mm]\nonumber
&&\times\quad
\int_{-\tau(E)/2}^{\tau(E)/2}\
e^{ir\left(E+\frac{\pi\eps^2}{\tau(E)}\right)/\eps^2}\
e^{iS(r)/\eps^2}\
\phi_0(A(r),B(r),\eps^2,a(r),\eta(r),x)\ dr\ \,+\ \,
O(\eps).
\eea
Here the orbit's energy $E$ is chosen to satisfy the
Bohr--Sommerfeld condition that the classical
action $I(E)$ of the orbit of energy $E$
must have $\displaystyle\frac{I(E)}{2\pi\eps^2}$
equal to a non-negative integer.
We note that $I(E)/(2\pi)$ is the area of the region of
phase space surrounded by the orbit.
We denote the period of the orbit by
$\displaystyle\tau(E)\,=\,\frac{\partial I}{\partial E}(E)$.
The energy of the quasimode is
$\displaystyle
E_{{\z},n}=E+\frac{\pi\eps^2}{\tau(E)}\,+\,O(\eps^3)$.
We choose
$a(0)=a_0$, $\eta_0=0$, $A(0)=A_{\w}$,
$B(0)=B_{\w}$, $S(0)=0$, and $\theta=E'_{\z}(a_0)A_{\w}$,
where $a_0$ is the point near $a_{\w}$ where $E_{\z}(a_0)=E$.
The $a(r)$, $\eta(r)$ and $S(r)$ satisfy
(\ref{adot}), (\ref{etadot}), and (\ref{Sdot})
with ${\zz}={\z}$, and $r$ in place of $t$.
The $A(r)$ and $B(r)$ satisfy
\bea\nonumber
\dot{A}(r)&=&i\,B(r)
\,+\,2\,\alpha(E)\,\eta(r)\,(E_{\z}'(a(r))A(r)+i\eta(r)B(r)),
\\[2mm]\nonumber
\dot{B}(r)&=&i\,E_{\z}''(a(r))\,A(r)\,+\,
2\,i\,\alpha(E)\,E_{\z}'(a(r))\,(E_{\z}'(a(r))A(r)+i\eta(r)B(r)).
\eea
The quantity $\alpha(E)$ in these equations is~
$\displaystyle\alpha(E)\ =\ \frac{\tau'(E)}{2\,\tau(E)}$.

See \cite{hagrob1,rousse2,rousse3} for detailed discussions.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Electronic Transitions and Franck--Condon
Factors}\label{FCsection}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our main result is summarized by the following theorem.

\begin{thm}\label{main}
Let $\Psi(x,\,t,\,\eps)$ denote the solution to the
Schr\"odinger equation (\ref{SE}) with initial condition
$$
\Psi(x,\,t_0,\,\eps)\ =\
\phi_m(A_{\w},B_{\w},\eps^2,a_{\w},0,x)\ \Phi_{\w}(x),
$$
where~
$t_0\,<\,\inf\,\mbox{\rm supp}\,f(\,\cdot\,,\,\eps)$.

\noindent
Then for~
$t\,>\,\sup\,\mbox{\rm supp}\,f(\,\cdot\,,\,\eps)$,~
the component of $\Psi(x,\,t,\,\eps)$ that is a multiple of
$\Phi_{\z}(x)$ is
%\bea\nonumber
%&&\mu\
%e^{it_0(
%GH  I TOOK YOUR SUGGESTION (several times)
%GH  I REFORMATTED THE EQUATION TO BE ON ONE LINE, ALSO.
%GH  I'll mark future places with %GH **
%% %% anytime it appears, why not substitute E_{{\w},m} for this %% %%
%E_{\w}(a_{\w})+(m+1/2)\omega_{\w}\eps^2
%% %% -------------------------------------------------------- %% %%
$$\mu\ e^{it_0(
E_{{\w},m})/\eps^2}\
\sum_n\ G_{\eps}(n,\,m)\ FC_{\eps}(n,\,m)\
e^{-itE_{{\z},n}/\eps^2}\
\psi_{n,\eps}(x)
\ \Phi_{\z}(x)
%\\[3mm]\nonumber
%&&\hskip 10.7cm
\ \,+\ \,O(\mu^2)\ +\ O(\mu\,\eps),
$$
%\eea
where the Franck--Condon factor is~~
$FC_{\eps}(n,\,m)=
\langle\,\psi_{n,\eps}(x),\,
\phi_m(A_{\w},B_{\w},\eps^2,a_{\w},0,x)
\,\rangle.$

In the generic case where~ $E_{\z}'(a_{\w})\,\ne\, 0$,
% $a_{\z}\neq a_{\w}$~ and
\bea\nonumber\hskip -28pt
FC_{\eps}(n,\,m)&=&
\frac{\pi^{1/4}\ 2^{-(m-1)/2}\ \,\eps^{1/2}}
{\sqrt{m!\ \tau(E_{\z}(a_{\w}))\,|E_{\z}'(a_{\w})|\,A_{\w}}}\ \,
H_m\left(\,\frac{B_{\w}\,
\left( E_{{\z},n}-E_{\z}(a_{\w})-\frac{\pi\eps^2}
{\tau(E_{\z}(a_{\w}))}\right)}{|E'_{\z}(a_{\w})|\,\eps}\,\right)
\\[3mm]\label{FCformula}
&&%\hskip 8mm
\times\quad\exp\left(\ -\
\frac{B_{\w}^2\ \left( E_{{\z},n}-E_{\z}(a_{\w})-\frac{\pi\eps^2}
{\tau(E_{\z}(a_{\w}))}\right)^2}{2\,E'_{\z}(a_{\w})^2\,\eps^2}\
\right)\ \left(\ 1\ +\ O(\eps^p)\ \right),
\eea
%GH  LINE ADDED.
where $p>0$.\\
In the non-generic case where~ $E_{\z}'(a_{\w})= 0$,~ we assume~
$E_{\z}''(a_{\w})>0$.\\ In this case, if~ $n-m$~ is odd, then~
$FC_{\eps}(n,\,m)=0$,~
and if ~ $n-m$~ is even, then
\bea\label{samecenter}\hskip -18pt
FC_{\eps}(n,\,m)&=& 2^{-\frac{m+n-1}{2}}\
\left( r+\frac {1}{r} \right)^{-\frac{m+n+1}{2}}
\\[3mm]\nonumber
&&\times
\sum_{\begin{array}{c}\vspace{-2mm}
\mbox{\scriptsize $j=0$}\cr\vspace{-2mm}
\mbox{\scriptsize $m-j$}\
\mbox{\scriptsize\rm even}\cr
\mbox{\scriptsize $n-j$}\
\mbox{\scriptsize\rm even}
\end{array}}^{\min(n,m)}\
\frac{(-1)^{\frac{n-j}2}\sqrt{(n!)(m!)}\ 4^j}{j!}\
\left(r-\frac 1r\right)^{\frac{m+n}2-j}\,w(m-j)\ w(n-j),
\eea
where~ $\displaystyle r\,=\,\frac{\omega_{\w}}{\omega_{\z}}$~ and~
$\displaystyle w(p)\ =\ \left\{
\begin{array}{lcl}\displaystyle
1&\mbox{if}&p=0,\vspace{1mm}\cr
\displaystyle
\frac{2}{p\,((\frac{p}{2}-1)!)}&\mbox{if}&p=2,\,4,\,6,\cdots.
\end{array}\right.
$

\vskip 4mm
If~
$f(t,\,\eps)\ =\ \eps^q\ f_0\left( t/\eps^{2-q}\right)\
\cos\left(\omega\,t/\eps^2\right)$~
with~ $0\le q\le 2$,~ then
\bea\label{G}\hskip -28pt
G_{\eps}(n,\,m)&=&
-\ i\ \sqrt{\frac{\pi}{2}}\ X_{{\z}{\w}}(a_{\w})\
\left[\
\widehat{f}_0\left(
%GH **
%\frac{E_{\w}(a_{\w})+(m+\frac 12)\omega_{\w}\eps^2
\frac{E_{{\w},m}
-E_{{\z},n}(\eps)+\omega}{\eps^q}
\right)\right.
\\[3mm]\nonumber\hskip -28pt
&&\hskip 45mm +\quad\left.
\widehat{f}_0\left(
%GH **
%\frac{E_{\w}(a_{\w})+(m+\frac 12)\omega_{\w}\eps^2
\frac{E_{{\w},m}
-E_{{\z},n}(\eps)-\omega}{\eps^q}
\right)\ \right],
\eea
where\quad
$X_{{\z}{\w}}(a_{\w})\ =\ \langle\,\Phi_{\z}(a_{\w}),\,
X\,\Phi_{\w}(a_{\w})\,
\rangle_{{\cal H}_{\mbox{\scriptsize\rm el}}}$.
\end{thm}

\vskip 5mm \noindent
{\bf Remarks.}\\
{\bf 1.}\quad
In physically realistic situations, the
$X_{{\z}{\w}}(x)$ is calculated from an electric dipole
moment of the molecule.\\
{\bf 2.}\quad
If $f_0$ is very smooth and $q>0$,
then the quantity (\ref{G})
will only be large where the change in energy in the
transition is $\pm\,\omega$. This corresponds
intuitively to the absorption or emission of one photon.
On the time scale that we are using, the laser frequency
is $\omega/\eps^2$, and the effective Planck constant is
$\eps^2$, so intuitively, a laser photon has energy $\omega$.\\
{\bf 3.}\quad
We shall see that $FC_{\eps}(n,\,m)$
is significantly large only when
$E_{{\z},n}-E_{\z}(a_{\w})\,=\,O(\eps)$. Since level spacings
in the $E_{\z}$ well are $O(\eps^2)$, the number of levels
that get significantly excited is $O(\eps^{-1})$.\\
{\bf 4.}\quad
Intuitively, for small $\eps$, each approximate nuclear
bound state $\psi_{n,\eps}$ or $\phi_m$ is concentrated in
an $O(\eps)$ neighborhood of a classical path in phase space.
For the Franck--Condon factor to be significantly large, the
two classical paths must have some places in phase space
where they are within an $O(\eps)$ distance of one another.
For $\phi_m$, the path is an ellipse with major axes on the
order of $\sqrt{m}\,\eps$,
so the place where the paths are close to
one another must be near $a_{\w}$, and
near a turning point of the path for~
$\psi_{n,\eps}$~ if~ $E'_{\z}(a_{\w})\ne 0$.\\
{\bf 5.}\quad
When $m>0$,~ $|FC_{\eps}(n,\,m)|$ can fluctuate
and have zeros in $n$ near the
values where $G_{\eps}(n,\,m)$ is large. Also, our formula
(\ref{FCformula})
predicts that $|FC_{\eps}(n,\,m)|$ has two largest peaks
with a energy separation of order~ $\sqrt{m}\ \eps$.\\
{\bf 6.}\quad
%GH  I REWORDED THIS SUBSTANTIALLY.
%When $a_{\w}=a_0$,
If $a_{\z}=a_{\w}$, where $a_{\z}$ is a local minimum of
$E_{\z}$ with $E''_{\z}(a_{\z})>0$,
%
%GH END THIS REWORDING
%
the formulas for $FC_\eps(n,\,m)$ are
just simplifications of the result of Proposition 7 of
\cite{hagrob1} that result from choosing the $A$ and $B$ to
be real. This yields formula (\ref{samecenter}).\\
{\bf 7.}\quad
%GH  I CHANGED THIS EQUATION ALSO AS IN PREVIOUS CHANGE.
If $a_{\z}=a_{\w}$
and
$E''_{\w}(a_{\w})=E''_{\z}(a_{\w})$, then
formula (\ref{samecenter}) has only one non-zero term, and
$FC(n,\,m)=\delta_{n,m}$.


\vskip 4mm
To prove Theorem \ref{main}, we begin by solving
the Schr\"odinger equation to zeroth order in $\eps$
and first order in $\mu$.

The leading term has order~ $\eps^0\,\mu^0$. It equals
$$
%\Psi_0(t)\ \,=\ \,
e^{i\,S_{\w}(t)/\eps^2}\
\phi_m(A_{\w}(t),B_{\w}(t),\eps^2,a_{\w},0,x)\ \Phi_{\w}(x),
$$
We note that\ \,$S_{\w}(t)\,=\,-\,(t-t_0)\,
E_{{\w},m}$,\ \,
$A_{\w}(t)\,=\,e^{i\omega_{\w}(t-t_0)}/\omega_{\w}$\ \,
and\ \,
$B_{\w}(t)\,=\,\omega_{\w}\,e^{i\omega_{\w}(t-t_0)}$\ \,
so that\ \,
$B_{\w}(t)/A_{\w}(t)\,=\,\omega_{\w}^2$\ \,is constant.

\vskip 5mm
The order~ $\eps^0\,\mu^1~$ term equals
\bea\nonumber\hskip -39pt
&&-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
U_{BO}(t,\,s)\ X\ f(s,\,\eps)\ U_{BO}(s,t_0)\ \psi_0(t_0)\ ds
\\[3mm]\nonumber\hskip -39pt
&=& -\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s,\,\eps)\ U_{BO}(t,\,s)\
e^{i\,S_{\w}(s)/\eps^2}\,
\phi_m(A_{\w}(s),B_{\w}(s),\eps^2,a_{\w},0,x)\
X\ \Phi_{\w}(x)\ ds
\\[3mm]\nonumber\hskip -39pt
&=& -\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s,\,\eps)\ U_{BO}(t,\,s)\
e^{i\,S_{\w}(s)/\eps^2}\,
\phi_m(A_{\w}(s),B_{\w}(s),\eps^2,a_{\w},0,x)
\\[3mm]\nonumber\hskip -39pt
&&\ \times\
\left[\Phi_{\w}(x)\,
\langle\,\Phi_{\w}(x),\,X\,\Phi_{\w}(x)\,
\rangle_{{\cal H}_{\mbox{\scriptsize\rm el}}}
+\,\Phi_{\z}(x)\,
\langle\,\Phi_{\z}(x),\,X\,\Phi_{\w}(x)\,
\rangle_{{\cal H}_{\mbox{\scriptsize\rm el}}}
+\,P_{\perp}(x)\,X\,\Phi_{\w}(x)
\right]\,ds.
\eea

\vskip 3mm \noindent
Here $P_{\perp}(x)$ denotes the projection in the electronic
Hilbert space onto the subspace orthogonal to
$\Phi_{\w}(x)$ and $\Phi_{\z}(x)$.

\noindent
We now define\quad
$X_{{\w}{\w}}(x)\,=\,
\langle\,\Phi_{\w}(x),\,X\,\Phi_{\w}(x)\,
\rangle_{{\cal H}_{\mbox{\scriptsize\rm el}}}$\quad
and\quad
$X_{{\z}{\w}}(x)\,=\,
\langle\,\Phi_{\z}(x),\,X\,\Phi_{\w}(x)\,
\rangle_{{\cal H}_{\mbox{\scriptsize\rm el}}}$.\\
Then our last expression can be written as
\bea\label{OK}\hskip -32pt
&&-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s,\,\eps)\ U_{BO}(t,\,s)\
e^{i\,S_{\w}(s)/\eps^2}\,\phi_m(A_{\w}(s),B_{\w}(s),\eps^2,a_{\w},0,x)
\\[4mm]\nonumber\hskip -32pt
&&\hskip 4cm \times\quad
\left[\ X_{{\w}{\w}}(x)\ \Phi_{\w}(x)\  
+\ X_{{\z}{\w}}(x)\ \Phi_{\z}(x)\
+\ P_{\perp}(x)\ X\ \Phi_{\w}(x)\
\right]\ ds
\\[4mm]\label{TS0}\hskip -32mm
&=&-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s,\,\eps)\ U_{BO}(t,\,s)\
e^{i\,S_{\w}(s)/\eps^2}\,\phi_m(A_{\w}(s),B_{\w}(s),\eps^2,a_{\w},0,x)
\\[4mm]\nonumber\hskip -32mm
&&\hskip 18mm\times\quad
\left[\ X_{{\w}{\w}}(a_{\w})\ \Phi_{\w}(x)\  
+\ X_{{\z}{\w}}(a_{\w})\ \Phi_{\z}(x)\
+\ P_{\perp}(x)\ X\ \Phi_{\w}(x)\
+\ O(\eps)\
\right]\ ds.
\eea
We concentrate on the components of this expression
that are in the $\Phi_{\w}(x)$ and $\Phi_{\z}(x)$
directions in the electronic Hilbert space. To leading order in
$\eps$, they are
\bea\label{muTerm}
&&\hskip -45pt
-\ i\ \frac{\mu}{\eps^2}\ X_{{\w}{\w}}(a_{\w})\
e^{i\,S_{\w}(t)/\eps^2}\,
\phi_m(A_{\w}(t),B_{\w}(t),\eps^2,a_{\w},0,x)\
\Phi_{\w}(x)\  \int_{t_0}^t\ f(s,\,\eps)\ ds
%}
\\[3mm]\nonumber
%\vskip 7mm
%\lefteqn{
&&\hskip -45pt
-\ i\ \frac{\mu}{\eps^2}\ X_{{\z}{\w}}(a_{\w})\ \Phi_{\z}(x)\
\int_{t_0}^t\ f(s,\,\eps)\
e^{i\,S_{\z}(t,s)/\eps^2}\,
\phi_m(A_{\z}(t,s),B_{\z}(t,s),\eps^2,
a_{\z}(t,s),\eta_{\z}(t,s),x)\ ds
\eea
In this expression, $a_{\z}(t,s)$,~$\eta_{\z}(t,s)$,~ etc.
are determined by using the dynamics associated with level
$E_{\w}$ for times between $t_0$ and $s$, and then the dynamics
associated with the $E_{\z}$ level for times between $s$ and $t$.
Intuitively, the vertical transition takes place at time $s$.

\vskip 3mm \noindent
{\bf Remarks.}\\
{\bf 1.}\quad
This last step is where the errors occur in \cite{Jilcott1,Jilcott2}.
Although it is buried in the notation, \cite{Jilcott1}
and \cite{Jilcott2}
have $X_{{\w}{\w}}(x)$ and $X_{{\z}{\w}}(x)$ outside the integrals
instead of $X_{{\w}{\w}}(a_{\w})$ and $X_{{\z}{\w}}(a_{\w})$.
These papers inadvertently
commute the $U_{BO}(t,s)$ past the multiplication operators
$X_{{\w}{\w}}(x)$ and $X_{{\z}{\w}}(x)$.
Without correction, these errors yield
an incorrect factor for the transition
amplitude in the second line of (\ref{muTerm})
in generic situations, {\it i.e.},
when $\eta_{\z}(t)$ is not identically zero and $X_{{\z}{\w}}(x)$
is not constant.\\
{\bf 2.}\quad
We have been somewhat careless about the error term
near the end of formula (\ref{TS0}). For
$f(t,\,\eps)\,=\,\eps^{q}\,f_0\left(\,t/\eps^{2-q}\,\right)
\,\cos(\omega\,t/\eps^2)$,
the rapid oscillations of the integrands yield
$O(\mu)$ results in (\ref{TS0}), despite the presence of the factor
of $\eps^{-2}$. By including more Taylor series terms
in going from (\ref{OK}) to (\ref{TS0}), one can show
that our neglecting the error term in (\ref{TS0}) produces
corrections to (\ref{muTerm}) that are $O(\mu\,\eps)$.
In several expressions below, we are similarly imprecise to avoid
uninteresting
%GH  THERE ARE SEVERAL PLACES IN SECTION 4 WHERE WE DO THIS, TOO.
%% %% only 1 l I think %% %%
technicalities.\\
%% %% ---------------- %% %%
{\bf 3.}\quad
If $f(t,\,s)\,=\,\eps^{q}\,f_0\left(\,t/\eps^{2-q}\,\right)
\,\cos(\omega\,t/\eps^2)$,
with $f_0$ smooth and $\omega\ne 0$, the first term in
(\ref{muTerm}) is very small as $\eps$ goes to zero when
$t\,>\,\sup\,\mbox{supp}\,f_0$, since the Fourier transform
of $f_0$ decays rapidly.


\vskip 4mm
Next, we expand the second term of (\ref{muTerm}) in terms
of the nuclear vibrational states of the upper electronic level.
We approximate the expansion coefficients by taking the inner product
of (\ref{muTerm}) with the leading order quasimodes
$\psi_{n,\eps}(x)\,\Phi_{\z}(x)$.

The electronic part of the inner product is
trivial, so we just obtain a nuclear variable inner product:

\vskip 4mm
\lefteqn{
\langle\,\psi_{n,\eps}(x)\,\Phi_{\z}(x),\
(\,\mbox{Formula (\ref{muTerm})}\,)\
\rangle_{\mbox{\scriptsize all variables}}
}

\vskip 5mm
\lefteqn{
=\quad -\ i\ \frac{\mu}{\eps^2}\ X_{{\z}{\w}}(a_{\w})
}

\vskip 5mm
\lefteqn{
\phantom{=\quad}\quad\times\quad\int_{t_0}^t\ f(s,\,\eps)\
\langle\,\psi_{n,\eps}(x),\
e^{i\,S_{\z}(t,s)/\eps^2}\,
\phi_m(A_{\z}(t,s),B_{\z}(t,s),\eps^2,a_{\z}(t,s),\eta_{\z}(t,s),x)\
\rangle\ ds
}

\vskip 5mm
\lefteqn{\phantom{=\quad}\quad +\quad O(\mu\,\eps)}

\vskip 5mm
\lefteqn{
=\quad -\ i\ \frac{\mu}{\eps^2}\ X_{{\z}{\w}}(a_{\w})
}

\vskip 5mm
\lefteqn{
\phantom{=\quad}\quad\times\quad\int_{t_0}^t\ f(s,\,\eps)\
\langle\,
\psi_{n,\eps}(x),\
e^{-i(t-s)
H_{\z}(\eps)/\eps^2}
\ e^{i\,S_{\w}(s)/\eps^2}\,
\phi_m(A_{\w}(s),B_{\w}(s),\eps^2,a_{\w},0,x)\
\rangle\ ds
}

\vskip 5mm
\lefteqn{\phantom{=\quad}\quad +\quad O(\mu\,\eps)}

\noindent where
$H_{\z}(\eps)\ =\
-\,\frac{\eps^4}{2}\,\frac{\partial^2}{\partial x^2}
\,+\,E_{\z}(x)$
has approximate eigenvector $\psi_{n,\eps}$ associated with $E_{{\z},n}$.

Hence, using the explicit formulas $A_{\w}(s)=A_{\w}\,e^{i\,\omega\,(s-t_0)}$,
$B_{\w}(s)=B_{\w}\,e^{i\,\omega\,(s-t_0)}$ and the definition (\ref{gaussians1D}),

\vskip 4mm
\lefteqn{
\langle\,\psi_{n,\eps}(x)\,\Phi_{\z}(x),\
(\,\mbox{Formula (\ref{muTerm})}\,)\
\rangle_{\mbox{\scriptsize all variables}}
}

\vskip 5mm
\lefteqn{=\quad
-\ i\ \mu\ X_{{\z}{\w}}(a_{\w})\
e^{-i(t-t_0)E_{{\z},n}/\eps^2}\
\langle\,\psi_{n,\eps}(x),\
\phi_m(A_{\w}(t_0),B_{\w}(t_0),\eps^2,a_{\w},0,x)\
\rangle\
}

\vskip 5mm
\lefteqn{
\hskip 18mm
\phantom{=\quad}\quad\times\quad\int_{t_0}^t\ f(s,\,\eps)\
e^{i(s-t_0)(E_{{\z},n}(\eps)-E_{\w}(a_{\w}))/\eps^2}\
%\left(\,\frac{A(0)}{A(s)}\,\right)^{m+1/2}\
e^{-i(s-t_0)(m+1/2)\omega_{\w}}\
\frac{ds}{\eps^2}
\quad +\quad O(\mu\,\eps).
}

% \vskip 3mm
% Note that in the last step, we have used
% (\ref{gaussians1D}) and the explicit formulas\\
% $A_{\w}(s)=A_{\w}\,e^{i\,\omega\,(s-t_0)}$\quad and\quad
% $B_{\w}(s)=B_{\w}\,e^{i\,\omega\,(s-t_0)}$.

Thus,
\bea\nonumber\hskip -32pt &&\hskip -25pt
\langle\,\psi_{n,\eps}(x)\,\Phi_{\z}(x),\
(\,\mbox{Formula (\ref{muTerm})}\,)\
\rangle_{\mbox{\scriptsize all variables}}
\\[3mm]\nonumber\hskip -32pt
&=&-\ i\ \mu\ X_{{\z}{\w}}(a_{\w})\
e^{-i(t-t_0)E_{{\z},n}/\eps^2}\
FC_{\eps}(n,\,m)\\[3mm]\nonumber
&&\hskip 4cm \times\quad\int_{t_0}^t\ f(s,\,\eps)\
e^{i(s-t_0)(E_{{\z},n}(\eps)-
%GH **
%E_{\w}(a_{\w})-(m+1/2)\omega_{\w}\eps^2)/\eps^2}
E_{{\w},m})/\eps^2}
\ \frac{ds}{\eps^2}\\[5mm]\label{dog}\hskip -32pt
&&+\quad O(\mu\,\eps),
\eea
where\quad
$FC_{\eps}(n,\,m)\ =\ \langle\,
\psi_{n,\eps}(x)
,\ \phi_m(A_{\w},B_{\w},\eps^2,a_{\w},0,x)\ \rangle$.

\vskip 5mm \noindent
{\bf Remark.}\quad
If we choose $f(t,\,\eps)\,=\,
\eps^{-q}\,f_0\left(\,t/\eps^{2-q}\,\right)\,
\cos\left(\omega\,t/\eps^2\right)$
with $0\le q\le 2$, we can say more about our last
expression for the transition amplitude. We assume
$f_0$ is a smooth function of compact support, and we look
at our final expression for~
$t_0<\inf\,\mbox{supp}\,f(\,\cdot\,,\,\eps)$~ and\\
$t>\sup\,\mbox{supp}\,f(\,\cdot\,,\,\eps)$.
Asymptotically for small $\eps$, we obtain
\bea\nonumber\hskip -25pt &&\hskip -25pt
-\ i\ \sqrt{\pi/2}\ \mu\ X_{{\z}{\w}}(a_{\w})\
e^{-itE_{{\z},n}/\eps^2}\
%GH **
%GH I ALSO REFORMATTED THE EQUATION.
%e^{it_0(E_{\w}(a_{\w})+(m+1/2)\omega_{\w}\eps^2)/\eps^2}\
e^{it_0E_{{\w},m}/\eps^2}\
FC_{\eps}(n,\,m)
\\[5mm]\label{hooray}\hskip -32pt
&&\quad\times\ \left[\
\widehat{f}_0\left(
%GH **
%\frac{E_{\w}(a_{\w})+(m+\frac 12)\omega_{\w}\eps^2
\frac{E_{{\w},m}
-E_{{\z},n}(\eps)+\omega}{\eps^q}\right)\right.
%\\[3mm]\nonumber\hskip -32pt
%&&\hskip 63mm
\ +\ \left.
\widehat{f}_0\left(
%GH **
%\frac{E_{\w}(a_{\w})+(m+\frac 12)\omega_{\w}\eps^2
\frac{E_{{\w},m}
-E_{{\z},n}(\eps)-\omega}{\eps^q}
\right)\ \right].
\eea
This proves formula (\ref{G}) of Theorem \ref{main}.\\

\vskip 4mm
We now give the leading order expression in $\eps$ for
$FC_{\eps}(n,\,m)$.
In this discussion, estimates are not uniform in $m$,
so we assume it is fixed.

Using (\ref{psin})
to represent $\psi_{n,\,\eps}(x)$,
we write $FC_{\eps}(n,\,m)$ as
$$\pi^{-1/4}\,\eps^{-1/2}\,
\sqrt{\frac{|\theta|}{2\,\tau(E)}}\
\int_{-\tau(E)/2}^{\tau(E)/2}\
e^{i\left(S(r)+rE+r\frac{\pi\eps^2}{\tau(E)}\right)/\eps^2}\
g(r,\,\eps)\ dr\,\ +\,\ O(\eps),$$
where
$$
g(r,\,\eps)\ =\
\langle\,\phi_0(A(r),B(r),\eps^2,a(r),\eta(r),x),\
\phi_m(A_{\w},B_{\w},\eps^2,a_{\w},0,x)\,\rangle.$$

The following lemma implies that $g(r,\,\eps)$ is large
only when $(a(r)-a_{\w})$ and $\eta(r)$ are $O(\eps^2)$.

\begin{lem}\label{phasespace}
Suppose the pair $(A(r),\,B(r))$ is restricted to a
compact set on which\\
$\mbox{\rm Re}\ \left(\,\overline{B(r)}\,A(r)\,\right)
\,=\,1$ is satisfied.
Then
$$
%\be\label{DistBound}
|g(r,\,\eps)|\ \le\
C_1\,\exp\left(-\
\frac{C_2\,\left(\,(a(r)-a_{\w})^2+\eta(r)^2\,\right)}
{\eps^2}\right),
%\ee
$$
where $C_1>0$ and $C_2>0$.
\end{lem}

\vskip 4mm \noindent
{\it Proof}:\ \, This is essentially contained in
Propositions 4 and 7 of \cite{hagrob1}.\ep

\vskip 4mm
It follows from this lemma that if
$E_{{\z},n}-E_{\z}(a_{\w})\gg\eps^2$, then
$FC_{\eps}(n,\,m)$ is bounded by
$\exp(-c/\eps^p)$ for some $c>0$ and $p>0$.
So, we choose $E_{{\z},n}$ near $E_{\z}(a_{\w})$
in order for $FC_{\eps}(n,\,m)$ to be significantly large.
This choice restricts
$a_0$ to a small compact neighborhood of $a_{\w}$.
It follows that for all
orbits we choose, $A(r)$ and $B(r)$ will lie in compact sets
as in Lemma \ref{phasespace}.
So, the only significant contribution
to the integral comes from values of $r$ that satisfy
$|r|<\eps^{1-\delta}$, where $\delta>0$ is arbitrarily small.

For such values of $r$, we have
\bea\label{asya}
a(r)&=&a_0\ -\ E'_{\z}(a_0)\,r^2/2\ +\ O(\eps^{3-3\delta}),
\\[3mm]\label{asyeta}
\eta(r)&=&-\,E'_{\z}(a_0)\,r\ +\ O(\eps^{2-2\delta}),
\\[3mm]\label{asyA}
A(r)&=&A_{\w}\ +\ i\,r\,B_{\w}\ +\ O(\eps^{2-2\delta}),
\\[3mm]\label{asyB}
B(r)&=&B_{\w}\ +\ i\,r\,\Big[E''_{\z}(a_0)\,+\,
2\,\alpha(E)\,(E'_{\z}(a_0))^2\Big]\,A_{\w}\
+\ O(\eps^{2-2\delta}),
%GH   I DECIDED TO REMOVE THE WORD "and" WHICH LOOKED WEIRD.
%\qquad\qquad\mbox{and}
\\[3mm]\label{asyS}
S(r)&=&-\,r\,E_{\z}(a_0)\ +\ O(\eps^{3-3\delta}).
\eea
We now use the formulas from Propositions 4 and 7 of
\cite{hagrob1} to write

\vskip 4mm
\lefteqn{
g(r,\,\eps)\ \,=\ \,
\frac{2^{-(m-1)/2}}{\sqrt{m!}}\ \,
\left(\,
\overline{B(r)}\,A_{\w}\,+\,\overline{A(r)}\,B_{\w}
\,\right)^{-(m+1)/2}\
\left(\,
\overline{B(r)}\,A_{\w}\,-\,\overline{A(r)}\,B_{\w}
\,\right)^{m/2}}

\vskip 6mm
\lefteqn{\hskip 15mm
\times\quad
H_m\left(\ \eps^{-1}\ \,
\frac{\overline{B(r)}(a(r)-a_{\w})\,
-\,i\,\overline{A(r)}\,\eta(r)}
{\sqrt{
(\overline{B(r)}\,A_{\w}\,-\,\overline{A(r)}\,B_{\w})\
(\overline{B(r)}\,A_{\w}\,+\,\overline{A(r)}\,B_{\w})
}}
\ \right)
}

\vskip 6mm
\lefteqn{\hskip 15mm
\times\quad\exp\left(\,-\
\frac{\overline{A(r)}\,A_{\w}\,\eta(r)^2+
\overline{B(r)}\,B_{\w}\,(a(r)-a_{\w})^2}
{2\ (\overline{B(r)}\,A_{\w}\,+\,\overline{A(r)}\,B_{\w})\
\eps^2}\,
+\,i\,\frac
{\overline{A(r)}\,B_{\w}\,\eta(r)\,(a(r)-a_{\w})}
{(\overline{B(r)}\,A_{\w}\,+\,\overline{A(r)}\,B_{\w})\
\eps^2}\,\right).}

\vskip 7mm
\begin{lem}For $|r|<\eps^{1-\delta}$, and
$|a_0-a_{\w}|<\eps^{1-\delta}$, with
$\delta<1/(m+1)$, we have

\vskip 6mm
\lefteqn{
\left(\
\frac{\overline{B(r)}\,A_{\w}\,-\,\overline{A(r)}\,B_{\w}}
{\overline{B(r)}\,A_{\w}\,+\,\overline{A(r)}\,B_{\w}}\
\right)^{m/2}\
H_m\left(\ \eps^{-1}\ \,
\frac{\overline{B(r)}(a(r)-a_{\w})\,
-\,i\,\overline{A(r)}\,\eta(r)}
{\sqrt{
(\overline{B(r)}\,A_{\w}\,-\,\overline{A(r)}\,B_{\w})\
(\overline{B(r)}\,A_{\w}\,+\,\overline{A(r)}\,B_{\w})
}}
\ \right)
}

\vskip 7mm
\lefteqn{\hskip 44mm =\quad
\left(\ \eps^{-1}\
(\overline{B(r)}(a(r)-a_{\w})\,
-\,i\,\overline{A(r)}\,\eta(r))
\,\right)^m\quad +\quad
O\left(\,\eps^{1-(m+1)\delta}\,\right).
}
\end{lem}

\vskip 4mm \noindent
{\it Proof}:\quad
We write the Hermite polynomial
in the lemma as
$\displaystyle H_m(z)\,=\,\sum_{j=0}^m\,c_j\,z^j$.
The $j=m$ term is
\bea\nonumber
&&\frac{2^m}
{(\overline{B(r)}\,A_{\w}\,
+\,\overline{A(r)}\,B_{\w})^m}\
\left(\ \eps^{-1}\
(\overline{B(r)}(a(r)-a_{\w})\,
-\,i\,\overline{A(r)}\,\eta(t))
\,\right)^m\\[3mm]\nonumber
&=&\left(\ \eps^{-1}\
(\overline{B(r)}(a(r)-a_{\w})\,
-\,i\,\overline{A(r)}\,\eta(r))
\,\right)^m\quad +\quad
O\left(\,\eps^{1-(m+1)\delta}\,\right).
\eea
This yields the expression displayed
explicitly on the right hand side in the lemma.
Another error term comes from the remaining terms
in the polynomial.

For $j<m$,
we use the asymptotics (\ref{asya})--(\ref{asyB})
and the fact that both $A_{\w}$ and $B_{\w}$ are real.
After some calculation, we see that the sum of
these terms equals
$$
\sum_{j=0}^{m-1}\ 2^{-\frac{m+j}2}\
(1\,+\,O(r))\
c_j\ (K\,r\,+\,O(r^2))^{\frac{m-j}2}
\left(\,\frac{B_{\w}\,(a_0-a_{\w})\,+\,O(r)}{\eps}\,\right)^j,
$$
where $K\,=\,i\,\Big(B_{\w}^2-\Big[E_{\z}''(a_0)+
2\alpha(E)(E'_{\z}(a_0))^2\Big]A_{\w}^2\Big)$.

We now simply estimate each term in this sum
when $|r|<\eps^{1-\delta}$ and
$|a_0-a_{\w}|<\eps^{1-\delta}$. For small $\eps$,
the largest term has $j=m-2$, since $c_{m-1}=0$.
This term equals
$$2^{-m+1}\,c_{m-2}\,(K\,r\,+\,O(r^2))\,
\left(\frac{B_{\w}(a_0-a_{\w})\,+\,O(r)}{\eps}\right)^{m-2}
\ =\ 2^{-m+1}\,c_{m-2}\,O(\eps^{1-\delta})\,
O(\eps^{-(m-2)\delta}).$$
This implies the lemma.\ep

\vskip 5mm \noindent
With this lemma, $B_{\w}\,A_{\w}=1$,
and (\ref{asya})--(\ref{asyB}), some
tedious, but straightforward calculations show that for
small $\delta$ and $|r|<\eps^{1-\delta}$,
\bea\label{june27a}\hskip -30pt
g(r,\,\eps)&=&
\left(\,\frac{2^{-m}\ \eps^{-m}}{\sqrt{m!}}\
\Big(\,B_{\w}\,(a_0-a_{\w})\,+\,
i\,A_{\w}\,E_{\z}'(a_0)\,r\,\Big)^m
+\,O\left(\,\eps^{p}\,\right)\,\right)
\\[3mm]\nonumber
&&\times\ \,
\exp\left(\,-\
\frac{B_{\w}^2\,(a_0-a_{\w})^2}{4\,\eps^2}\,-\,
\frac{A_{\w}^2\,(E_{\z}'(a_0))^2\,r^2}{4\,\eps^2}\,-\,
\frac{i\,A_{\w}\,B_{\w}\,E_{\z}'(a_0)\,(a_0-a_{\w})\,r}{2\,\eps^2}
\,\right).
\eea
where $p>0$
(and can be chosen arbitrarily close to 1 if $\delta$ is small enough).


We can now evaluate the leading asymptotics for
$FC_{\eps}(n,\,m)$. We first restrict the
$r$ integration region to $|r|<\eps^{1-\delta}$, then
substitute the above asymptotics for $g$. Then we
extend the integration to all $r$, again making an exponentially
small error. We can then evaluate the integral explicitly
%GH I COULD SCAN IN THE PAGE OF THE BOOK FOR YOU IF YOU LIKE.
%GH IT MIGHT ALSO BE AVAILABLE ON THE WEB.
%GH TRY A SEARCH ON GRADSHTEYN.
(\cite{GR} page 382, formula 3.462.4) to obtain
\bea\nonumber\hskip -21pt
FC_{\eps}(n,\,m)&=&
\frac{\pi^{1/4}\ 2^{-(m-1)/2}\ \,\eps^{1/2}}
{\sqrt{m!\ \tau(E)\,|E_{\z}'(a_0)|\,A_{\w}}}\ \,
H_m\left(\,\frac{B_{\w}\,(a_0-a_{\w})}{\eps}\,\right)\ \,
\exp\left(\ -\
\frac{B_{\w}^2\ (a_0-a_{\w})^2}{2\,\eps^2}\ \right).
\\[3mm]\label{june27b}
&&+\quad O\left(\eps^{1/2+p}\right).
\eea
To get the error estimate, here, we actually include some of
the error term from (\ref{june27a}) to get easy bounds.
We then observe that including these terms yields corrections
that are themselves $O\left(\eps^{1/2+p}\right)$.
These corrections are small because of oscillations in the
integral, and we have dropped them in (\ref{june27b}).

Expression (\ref{june27b}) is only significant if
$a_0-a_{\w}\,=\,O(\eps)$. This condition is equivalent to
$E_{{\z},n}-E_{\z}(a_{\w})\,=\,O(\eps)$. Since level spacings
in the $E_{\z}$ well are $O(\eps^2)$, the number of levels
that get significantly excited is $O(\eps^{-1})$.

Furthermore,  since~
$E_{{\z},n}=E_{\z}(a_0)+\frac{\pi\eps^2}{\tau(E)}\,+\,O(\eps^3)$
~and\\
$E_{\z}(a_0)-E_{\z}(a_{\w})=E'_{\z}(a_{\w})(a_0-a_{\w})+O((a_0-a_{\w})^2)$,
\bea\nonumber\hskip -1mm
FC_{\eps}(n,\,m)
&=&
\frac{\pi^{1/4}\ 2^{-(m-1)/2}\ \,\eps^{1/2}}
{\sqrt{m!\ \tau(E)\,|E_{\z}'(a_0)|\,A_{\w}}}\ \,
H_m\left(\,\frac{B_{\w}\,
\left( E_{{\z},n}-E_{\z}(a_{\w})-\frac{\pi\eps^2}
{\tau(E)}\right)}{|E'_{\z}(a_{\w})|\,\eps}\,\right)
\\[3mm]\nonumber
&&\hskip 14mm \times\quad\exp\left(\ -\
\frac{B_{\w}^2\ \left( E_{{\z},n}-E_{\z}(a_{\w})-\frac{\pi\eps^2}
{\tau(E)}\right)^2}{2\,E'_{\z}(a_{\w})^2\,\eps^2}\ \right)
\ \,  +\ \, O(\eps^{1/2+p}),
\eea
for some $p>0$.

\vskip 4mm
\noindent
{\bf Remarks}.\\
{\bf 1.}\quad
The Franck--Condon factors are small unless
$E=E_{\z}(a_0)\approx E_{{\z},n}\approx E_{\z}(a_{\w})$.
Since various quantities
depend continuously on the energy, we have
\bea\nonumber\hskip -3mm
FC_{\eps}(n,\,m)
&=&
\frac{\pi^{1/4}\ 2^{-(m-1)/2}\ \,\eps^{1/2}}
{\sqrt{m!\ \tau(E_{\z}(a_{\w}))\,|E_{\z}'(a_{\w})|\,A_{\w}}}\ \,
H_m\left(\,\frac{B_{\w}\,
\left( E_{{\z},n}-E_{\z}(a_{\w})-\frac{\pi\eps^2}
{\tau(E_{\z}(a_{\w}))}\right)}{|E'_{\z}(a_{\w})|\,\eps}\,\right)
\\[3mm]\nonumber\hskip -2mm
&&\hskip 8mm \times\quad\exp\left(\ -\
\frac{B_{\w}^2\ \left( E_{{\z},n}-E_{\z}(a_{\w})-\frac{\pi\eps^2}
{\tau(E_{\z}(a_{\w}))}\right)^2}{2\,E'_{\z}(a_{\w})^2\,\eps^2}\ \right)
\ \, +\ \, O(\eps^{1/2+p}),
\eea
when $E_{{\z},n}-E_{\z}(a_{\w})\,=\,O(\eps)$.\\[3mm]
{\bf 2.}\quad
The Franck--Condon factors arise from expanding a normalized
wave packet in an orthonormal basis.
So, if we sum their absolute squares over $n$,
we must get $1$. We can observe this from our formulas.
The Bohr--Sommerfeld conditions require~
$I(E_{{\z},n+1})-I(E_{{\z},n})\,\approx\,2\pi\eps^2$. Thus,~
$\displaystyle\frac{\partial I}{\partial E}\ \Delta E
\ \approx\ 2\pi\eps^2$,~
and hence,~
$\displaystyle\Delta E\ \approx\
\frac{2\pi\eps^2}{\tau(E)}$.
Therefore,
\bea\nonumber\hskip -20pt
\sum_n\,|FC_{\eps}(n,\,m)|^2&\approx&
\frac{1}{\Delta E}\
\int\ |FC_{\eps}(n({\cal E}),\,m)|^2\ d{\cal E}
\\[3mm]\nonumber
&\approx&
\frac{\tau(E)}{2\,\pi\,\eps^2}\ \,
\frac{\pi^{1/2}\ 2^{-(m-1)}\ \,\eps}
{m!\ \tau(E)\,|E_{\z}'(a_{\w})|\,A_{\w}}
\\[3mm]\nonumber
&&\times\quad\int\ H_m\left(\,
\frac{B_{\w}\,({\cal E}-E_{\z}(a_{\w}))}
{|E'_{\z}(a_{\w})|\,\eps}\,\right)^2\
\exp\left(\,-\
\frac{B_{\w}^2\,({\cal E}-E_{\z}(a_{\w}))^2}
{|E'_{\z}(a_{\w})|^2\,\eps^2}\,\right)\
d{\cal E}
\\[3mm]\nonumber
&=&
\frac{2^{-m}\ \pi^{-1/2}}{m!}\ \,
\int\ H_m(z)^2\ e^{-z^2}\ dz
\\[3mm]\nonumber
&=&1.
\eea

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Expansions to all orders in $\eps$ and $\mu$}
\label{jilcott}
\setcounter{equation}{0}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Since our main results do not depend on this section, we
do not provide many details that can be found elsewhere.
We restrict attention to the situation where the electron
Hamiltonian has finitely many levels that never cross one
another. A more general situation is
discussed in \cite{Jilcott1,Jilcott2}, but identifying and
correcting the error is the same.

The basic idea for correcting the results (Theorems 3.2 and 3.3)
of \cite{Jilcott2} is to
redefine and take superpositions of the functions
$\Psi_{(i,k_1,\dots,k_m),n}(\eps,s_1,\dots,s_m,t)$ that appear
in these theorems. Their definitions are described near the end
of Section 3.2 of \cite{Jilcott2}, but they are not written out
explicitly. However,
the dependence of these functions on the multi-index $j$ for
the semiclassical wave packets is suppressed in
\cite{Jilcott1,Jilcott2}. When the correction is made, we
cannot omit this dependence because there is some mixing
of different values of $j$.

Throughout this discussion, the value of $n\ge 0$ is fixed.
We do all calculations modulo $\eps^{n+1}$ errors and perform
a perturbation expansion in powers of $\mu$.

We begin this discussion with some preliminary results.
There are explicit constructions of asymptotic expansions
for solutions to (\ref{SE}) when $\mu=0$ that can be found in
\cite{tdbo,texas,memoir}.
For any electronic level $i$ and any multi-index $j$,
the constructions
in \cite{tdbo,texas,memoir} yield a unique approximate
solution $\psi_{n,i,j}(x,\,t,\,s)$
to (\ref{SE}) with $\mu=0$, subject to the condition that
$$
\langle\,\Phi_i(x),\,
\psi_{n,i,j}(x,\,s,\,s)\,
\rangle_{{\cal H}_{\mbox{\scriptsize el}}}
\ =\ e^{iS_i(s)/\eps^2}\,
\phi_j(A_i(s),B_i(s),\eps^2,a_i(s),\eta_i(s),x).
$$
The dynamics of $A_i(t)$, $B_i(t)$, $a_i(t)$, $\eta_i(t)$
and $S_i(t)$ are determined by the $i^{\mbox{\scriptsize th}}$
electronic level.

The function $\psi_{n,i,j}(x,\,t,\,s)$ is a
time--dependent finite sum of the form
$$
%GH  I CHANGED S TO S_i
\psi_{n,i,j}\ =\ e^{iS_i/\eps^2}\,\left(\,\Phi_i(x)\,\phi_j(x)\ +\
\sum_{p=1}^{n+2}\ \eps^p\
%GH  I ADDED THE ARGUMENT t TO f
\sum_{k,j'}\ f_{p,k,j'}(x,\,t)\,\Phi_k(x)\,\phi_{j'}(x)\,\right).
$$
%GH  I ADDED THE ARGUMENT t TO f
For each $t$, we approximate each $f_{p,k,j'}(x,\,t)$ by its
Taylor series in $(x-a(t))$ of order $\max\,\{n-p,\,0\}$.
This adds an $O(\eps^{n+1})$ error.
When this Taylor series multiplies $\phi_{j'}$, the
product can be written as a finite linear combination of more
$\phi_{j'}$, with $|j'-j|\le 3\,\min\{p,\,n\}+\max\,\{n-p,\,0\}$.
Since $p\le n+2$, this implies that up to an $O(\eps^{n+1})$
error $\psi_{n,i,j}(x,\,t,\,s)$ is a finite linear combination
of $\Phi_k(x)\,\phi_{j'}(x)$ with $|j'-j|\le 3n+4$.

Thus, modulo an $O(\eps^{n+1})$ error,
$\psi_{n,i,j}$ is a time--dependent finite sum of the form
\be\label{psi}
%GH  I CHANGED S TO S_i
\psi_{n,i,j}\ =\ e^{iS_i/\eps^2}\,\left(\,\Phi_i(x)\,\phi_j(x)\ +\
\sum_{p=1}^{n+2}\ \eps^p\
%GH  I ADDED THE ARGUMENT t TO c
\sum_{k,j'}\ c^{(0)}_{p,k,j'}(t)\,\Phi_k(x)\,\phi_{j'}(x)\,\right)\
+\ O(\eps^{n+1}).
\ee

\vskip 4mm
In addition to this result, our arguments require that we be able
to write any $\Phi_i(x)\,\phi_{j}(x)$
as a finite linear combination of the $\psi_{n,i'',j''}$
up to an $O(\eps^{n+1})$ error.
To see that this can be done, we iteratively make use of
expression (\ref{psi}). First, we rewrite (\ref{psi}) as
\bea\nonumber
%GH  I CHANGED S TO S_i
e^{-iS_i/\eps^2}\,\psi_{n,i,j}&=&\Phi_i(x)\,\phi_j(x)\ +\
%GH  I ADDED THE ARGUMENT t TO c
\eps\ \sum_{k,j'}\ c^{(0)}_{1,k,j'}(t)\,\Phi_k(x)\,\phi_{j'}(x)
\\[3mm]\nonumber
&&+\quad
\sum_{p=2}^{n+2}\ \eps^p\
%GH  I ADDED THE ARGUMENT t TO c
\sum_{k,j'}\ c^{(0)}_{p,k,j'}(t)\,\Phi_k(x)\,\phi_{j'}(x)\
+\ O(\eps^{n+1}).
\eea
Then we use (\ref{psi}) again to rewrite the order $\eps$
terms as linear combinations of $\psi_{n,k,j'}$ plus higher
order corrections. Thus, we see that
%% %% we should be careful with the S that stands for different dynamics %% %%
\bea\nonumber
%GH  I CHANGED S TO S_i
e^{-iS_i/\eps^2}\,\psi_{n,i,j}&=&\Phi_i(x)\,\phi_j(x)\ +\
%GH  I ADDED THE ARGUMENT t TO c
\eps\ \sum_{k,j'}\ c^{(0)}_{1,k,j'}(t)\,
%GH  I CHANGED S TO S_i
%VR  I think it is S_k                 CHANGED GH
e^{-iS_k/\eps^2}\,\psi_{n,k,j'}(x)
\\[3mm]\nonumber
&&+\quad
\sum_{p=2}^{n+2}\ \eps^p\
%GH  I ADDED THE ARGUMENT t TO c
\sum_{k,j'}\ c^{(1)}_{p,k,j'}(t)\,\Phi_k(x)\,\phi_{j'}(x)\
+\ O(\eps^{n+1}),
\eea
where the sums on the right hand side are finite. We now
repeat this procedure to replace the
$\Phi_k(x)\,\phi_{j'}(x)$ in the order $\eps^2$ terms by
$\psi_{n,k,j'}$ plus higher order corrections.

By repeating this process, we obtain
$$
%GH  I CHANGED S TO S_i
e^{-iS_i/\eps^2}\,\psi_{n,i,j}\ =\ \Phi_i(x)\,\phi_j(x)\ +\
\sum_{p=1}^{n+2}\ \eps^p\
%GH  I ADDED THE ARGUMENT t TO c
\sum_{k,j'}\ c^{(p-1)}_{p,k,j'}(t)\,
%GH  I CHANGED S TO S_i
%VR  I think it is S_k           CHANGED GH
e^{-iS_k/\eps^2}\,\psi_{n,k,j'}(x)\
+\ O(\eps^{n+1}),
$$
where the sums are finite.

It follows immediately that
\bea\nonumber
%GH  I CHANGED S TO S_i
%VR the leading order term should get S_i and the others S_k  CHANGED GH
\Phi_i(x)\,\phi_j(x)&=&\left(\,e^{-iS_i/\eps^2}\,
\psi_{n,i,j}\ -\
\sum_{p=1}^{n+2}\ \eps^p\
%GH  I ADDED THE ARGUMENT t TO c
\sum_{k,j'}\ c^{(p-1)}_{p,k,j'}(t)\,e^{-iS_k/\eps^2}\,
\psi_{n,k,j'}(x)\,\right)
\\[3mm]\label{finally}
&&+\ O(\eps^{n+1}),
\eea
where the sums are finite.


\vskip 4mm
With these preliminary facts established, we now turn
to the main point of this section.


We fix the order $n$ of the expansion in $\eps$ and the
choice of the $i^{\mbox{\scriptsize th}}$ initial electronic
level. We also fix
a value of the multi-index $j$ that labels an initial
semiclassical nuclear wave packet at time $t_0$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   ORDER 0 IN mu
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vskip 4mm \noindent
{\bf Order 0 in $\mu$.}\quad
To zeroth order in $\mu$ and $n^{\mbox{\scriptsize th}}$
order in $\eps$, we have the approximate solution
$$\Psi_{n,i,j}(\eps,t)\,=\,\psi_{n,i,j}(x,\,t,\,t_0).$$
This is exactly as in \cite{Jilcott1,Jilcott2}, although
our notation is slightly different.


%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%   ORDER 1 IN mu
%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vskip 4mm\noindent
{\bf First Order in $\mu$.}\quad
To calculate the higher order corrections in $\mu$,
we use the Dyson expansion of \cite{Jilcott1,Jilcott2}.
We let $U_{BO}^{(n)}(t,\,s_1)$ denote the
Born--Oppenheimer propagator with error $O(\eps^{n+1})$.
Then the first order term in $\mu$ is
\bea\nonumber
&&-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s_1,\,\eps)\ U^{(n)}_{BO}(t,\,s_1)\ X\
U^{(n)}_{BO}(s_1,t_0)\ \psi_{n,i,j}(x,\,t_0,\,t_0)\ ds_1
\\[5mm]\label{grog}
&=&-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s_1,\,\eps)\ U^{(n)}_{BO}(t,\,s_1)\ X\
\psi_{n,i,j}(x,\,s_1,\,t_0)\ ds.
\eea
We use (\ref{psi}) to rewrite
$\psi_{n,i,j}(x,\,s_1,\,t_0)$ as a finite linear
combination
$$\sum_{i',\ j'}\ \alpha_{i',j'}(s_1,\,\eps)\
\Phi_{i'}(x)\,\phi_{j'}(x)\ +\ O(\eps^{n+1}).$$
The operator $X$ applied to this yields another
finite linear combination
$$\sum_{i'',\ i',\ j'}\ \beta_{i'',i',j'}(s_1,\,\eps)\
\Phi_{i''}(x)\ X_{i'',i'}(x)\ \phi_{j'}(x)
\ +\ O(\eps^{n+1}),$$
where $X_{i'',i'}(x)\ =\
\langle\,\Phi_{i''},\,X\,\Phi_{i'}\,\rangle$.
We make another $O(\eps^{n+1})$ error by replacing
$X_{i'',i'}(x)$ by its $n^{\mbox{\scriptsize th}}$
order Taylor series.
$$
\sum_{|m|\le n}\
%VR the two a(.) should have the same argument and I think it should be s_1 
%because it is the time at which we apply the X     CHANGED GH
\frac{\left(D^mX_{i'',i'}\right)(a(s_1))\,(x-a(s_1))^m}{m!}.
$$
When this multiplies the $\phi_{j'}(x)$ we again obtain
a finite linear combination
$$\sum_{i'',\ j''}\ \gamma_{i'',j''}(s_1,\,\eps)\
\Phi_{i''}(x)\ \phi_{j''}(x)
\ +\ O(\eps^{n+1}).$$
We apply (\ref{finally}) to write this as
$$
X\,\psi_{n,i,j}(x,\,s_1,\,t_0)\ =\
\sum_{i_1,\ j_1}\ \zeta_{i_1,j_1}(\eps,\,s_1)\
\psi_{n,i_1,j_1}(x,\,s_1,\,s_1)
\ +\ O(\eps^{n+1}),
$$
where the linear combination is finite.

We use this result in (\ref{grog}) and then
do the Born--Oppenheimer propagation to obtain
\bea\nonumber
&&-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s_1,\,\eps)\ U^{(n)}_{BO}(t,\,s_1)\ X\
\psi_{n.i,j}(x,\,s_1,\,t_0)\ ds_1
\\[5mm]\label{TheEnd}
&=&
\sum_{i_1,\ j_1}\
-\ i\ \frac{\mu}{\eps^2}\ \int_{t_0}^t\
f(s_1,\,\eps)\ \zeta_{i_1,j_1}(\eps,\,s_1)
\psi_{n,i_1,j_1}(x,\,t,\,s_1)\ ds_1\ +\ O(\eps^{n+1}).
\eea
In order to get the proper error estimate here, we
again use the assumed form of $f(s_1,\,\eps)$ to
overcome the factor of~
$\displaystyle\frac{1}{\eps^2}$~ as in Section \ref{main}.

\vskip 4mm
Intuitively, the terms that have a given value
of $i_1$ have made a jump at time $s_1$ from level
$i$ to level $i_1$, after which, they propagate
according to the Born--Oppenheimer dynamics associated
with level $i_1$. The sum over $j_1$ results from
the nuclear wave packet changing somewhat when the jump
occurs. Thus, the precise ``vertical transition''
intuition is only accurate to leading order in $\eps$.

\vskip 4mm
We can go to higher orders in $\mu$ by treating
the higher order Dyson expansion terms in
\cite{Jilcott1,Jilcott2} by the same technique.
Where the operator $X$ occurs in the formal expressions,
we use (\ref{psi}) to rewrite everything as a finite linear
combination of $\Phi_{i'}(x)\,\phi_{j'}(x)$.
We then use Taylor series to expand
$X_{i'',i'}(x)$. Then we use (\ref{finally}) to reexpress
the result
in terms of new basis vectors $\psi_{n,i_k,j_k}$

The order $\mu^m$ term makes $m$ transitions
between electronic levels at times
$s_1,\,s_2,\,\cdots,\,s_m$ as expected. However, the
$O(\eps^n)$--accurate wave function is an enormous
sum of terms.

\vskip 4mm \noindent
{\bf Remark.}\quad
When $n=0$, these expressions are much simpler.
In particular, expression (\ref{TheEnd}) reduces to
$$\sum_{i_1}\
-\ i\ \frac{\mu}{\eps^2}\ \Phi_{i_1}(x)\ \int_{t_0}^t\
f(s_1,\,\eps)\ X_{i_1,i}(a_i(s_1))\
e^{iS_{i_1,i}(t,s_1)/\eps^2}\ \phi_j(x,\,t,\,s_1)\ ds_1\
+\ O(\eps).
$$
Here the semiclassical wave packet
$\phi_j(x,\,t,\,s_1)$ and the $S_{i_1,i}(t,s_1)$ propagate
according to the dynamics of level $i$ from time $t_0$ to
time $s_1$ and the dynamics of level $i_1$ from time $s_1$
to time $t$. This differs from \cite{Jilcott1,Jilcott2}
for $n=0$
only in the presence of the factor $X_{i_1,i}(a_i(s_1))$
in the integrand.

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\parskip 1.5mm
\baselineskip=17pt
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Elementary Processes of Photochemical Reactions.
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\bibitem{GR}Gradshteyn, I.S. and Ryzhik, I.M.:\,\
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\bibitem{tdbo}Hagedorn, G.A.:\,\
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\bibitem{texas}Hagedorn, G.A.:\,\
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\bibitem{Jilcott2}Jilcott, S.W.:\,\
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\bibitem{rousse2}V. Rousse:\,\
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\end{thebibliography}

\end{document} 
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